User’s Guide to the Free Speech Calculus Text
Version 0.5
W. Ethan Duckworth
August, 2005
Contents
1 Introduction
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2 Who are the users?
3
3 How a teacher or serious user uses the text
3.1 Inputting files . . . . . . . . . . . . . . . . . . . .
3.2 Pseudo-paragraphs . . . . . . . . . . . . . . . . .
3.3 Controlling pseudo-paragraphs with \Show . . . .
3.3.1 Examples of the Show command . . . . .
3.3.2 Caveat: the Show command requires keys
3.3.3 Fine-tuning with label . . . . . . . . . .
3.3.4 Other arguments to Show . . . . . . . . .
3.3.5 A special psuedo-paragraph: homework .
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4 How a Contributor uses the text
4.1 Getting your stuff into the main distribution . . . . .
4.2 Figuring out where to edit . . . . . . . . . . . . . . . .
4.3 What to write . . . . . . . . . . . . . . . . . . . . . . .
4.4 Writing stuff up . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Pseudo-paragraphs . . . . . . . . . . . . . . . .
4.4.2 Creating new pseudo-paragraph environments
key-values . . . . . . . . . . . . . . . . . . . . .
4.4.3 Pseudo-sections . . . . . . . . . . . . . . . . . .
4.4.4 Notation . . . . . . . . . . . . . . . . . . . . . .
4.5 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Formats and openness . . . . . . . . . . . . . .
4.5.2 Some other graphics tools . . . . . . . . . . . .
4.5.3 Graphic size . . . . . . . . . . . . . . . . . . . .
4.5.4 Graphics commands . . . . . . . . . . . . . . .
4.5.5 Future graphics procedures . . . . . . . . . . .
4.6 Some re-defined LATEX commands . . . . . . . . . . . .
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§1: Introduction
5 How a hacker uses the text
1
2
20
Introduction
How would you like to have a Calculus text that you could customize to the
way you like to teach Calculus? How would you like to have a text where you
can print out just the examples? Or just the definitions? Or suppress all the
examples? How would you like it if there could be an electronic version with
hyperlinks to definitions, hyperlinks for extra detail, hyperlinks to historical
information? How would you like it if this text was free in all the usual senses:
free as in speech (you can use it how you like, other people can use it how
they like) free as in open to change (you can add examples, pictures, historical
interludes to your version or submit them to the master version) free as in ice
cream (you can give it away for no charge to your students).
Well, I think the technology exists to make all these things and more possible. I do not claim that I have accomplished all of these goals yet. At present
(August 2005) we have about 400 pages of of text (culled from a variety of
sources), and graphics. This means that we need considerably more material to
be complete. I have implemented LATEX code which allows many options and
combinations in how the document can be processed. I have not added all of the
key-values (a type of meta-data for the material) for showing/hiding all combinations of examples, definitions, etc. I have not added any of the hyperlinks for
the electronic version, though this is fairly easy to do with standard packages
and using PDF format. (Note that the goal of widespread use of this text makes
the PDF format the best choice for the final product.)
This calculus text is copyrighted under the Gnu Free Documentation License
(GFDL). Loosely speaking here are the terms of this license:
• You are free to copy, redistribute, change, print, sell, and otherwise use in
any manner part or all of this document,
• Any work derived from this text must also be covered by the GFDL. (This
only applies to that part of your work derived from this text. It is up to
you whether those parts of your work which are not based on this text is
covered by the GFDL. Also, you can quote, with attribution and copyright
notice and subject to fair use provisions, from this text like you would from
any copyrighted work.)
• Anyone distributing works covered by the GFDL must provide source code
or other editable files for the material which is distributed. In the case of
this text that means the LATEX code, as well as the source documents for
creating the graphics.
• If you make changes to this text and redistribute them, you should name
the finished product something different than “The Free Speech Calculus
Text” or “The Free Speech Calculus Text: original version”. You may
§2: Who are the users?
3
choose derivative names like “The Free Speech Calculus Text: the John
Doe version”.
The material is a combination of independent works by W. Ethan Duckworth, Paul Garrett, Michael Livshits, and the various contributors to Wikibooks. All the independent works were licensed under the GFDL; W. Ethan
Duckworth combined them into one whole, added some material and LATEX code
(this is the type of thing that the GFDL encourages).
2
Who are the users?
If you think about it, an ordinary Calculus textbook has two types of users:
students and teachers. If we oversimplify we might say that the students mostly
look at the examples and the exercises. The teachers look at the ordering of
sections, what’s in each section, and the exercises.
The Free Speech Calculus Text will probably have four types of users.
The student. At present I think the student will only use material that the
teacher prepares for him or her. Once the text is more complete, here is
how I envision a student using it.
Firstly, the electronic version of the text can complement the printed version. In an electronic version there could be extra historical information,
there can be keyword searches, hyperlinks to definitions, etc.
In the future, I also imagine having a graphical interface for a student to be
able to choose to see only examples. Or to see all the examples using sin(x).
After they make their choices, TEX would run in the background (invisibly)
and they would have a sub-book. The graphical interface can actually be
quite simple, because the configuration required can be answered with a
small set of yes or no type questions. The answers to these questions would
be written out in a configuration file, and then TEX would run, using this
configuration.
The teacher (aka a serious user). At present, the material is not complete,
so I think a teacher would only use this text for a few extra examples, or
for a few sections, not for the whole class. Once this text is complete,
here’s how I envision a teacher using it.
In addition to the uses that teachers currently have for textbooks, I imagine a teacher using the Free Speech Calculus Text more interactively. If
the teacher wants to include more history, they could just say “show me
the history”. If they want to omit trigonometry or make it come late in
the class, they can choose to do this.
At present (more correctly, once the material in the text is complete), the
teacher can do this with a small amount of knowledge of LATEX. In the
future, I imagine various versions of the text that the typical teacher could
choose from, even with zero knowledge of LATEX. Farther in the future I
again imagine a graphical interface, as mentioned for the student user.
§3: How a teacher or serious user uses the text
4
Contributors. At present the text are still incomplete, and I welcome contributions. By all means, contribute any way you like. Take the text and
teach a class with it. Or extract some examples from them. Hand out
supplements, etc. Probably if you try to use the text you will see something you wish was there, but isn’t. This is the perfect thing to contribute
because you’re interested in it, and because there’s probably someone else
who wishes it was there too. If you do any of these things and want to
return something to the community, share what you’ve added, either by
posting it on the web, or by contacting me and having it folded back into
the main distribution of the Free Speech Calculus Text. You are also
welcome to submit even small changes, like grammatical corrections, or
inserting a sentence.
At present, I imagine these text as being like a moderated Wiki. By
this I mean that virtually any changes that are submitted to me will
be incorporated, and that I hope you submit complete changes, not just
suggestions. In fact, in the future, we might turn the text into some sort
of Wiki, or Wiki-hybrid; thus allowing anyone to make changes directly
without any intervention on my part.
A consequence of this policy will be, I hope, a rapidly growing text with
a large number of alternative approaches and extra material. (If you feel
that I accept too many/few contributions then, because the text is licensed
under the GFDL, you are welcome to copy the whole thing and create an
alternate project with different rules, but of course with the same license.)
For this reason, one of the main goals in writing the LATEX code has been
to make it easy to show/hide parts of the text. In the future, I imagine
there being more material than anyone would want in a single book and,
if this becomes the case, one of the things contributors could do is provide
different versions which show different subsets of the material. So there
could be a standard version, a friendly version, a formal version, a version
with extra history, a version based on infinitesimals, a version that doesn’t
use limits, etc. The tools for doing this are described below.
Hackers. Here I of course mean people who like fooling around with computers
and code, not malicious people who like to attack computers which belong
to others. I describe below some goals I would have below for this type of
user, though hackers are notoriously prone to doing what they like rather
than following directions! Good.
3
How a teacher or serious user uses the text
This is the actual “user guide”: what are the commands, how do they work,
etc. I am not going to cover LATEX basics, rather things that are particular to
this text.
§3: How a teacher or serious user uses the text
5
Figure 1: Simple master.tex file
\documentclass{article}
\usepackage{free_calc}
\begin{document}
\chapter{Derivatives}
\include{rules}
\include{derivative_applications}
\chapter{Antiderivatives}
\include{rules}
\include{integral_applicaitions}
\end{document}
3.1
Inputting files
The file free_calc.sty loads various packages and defines many commands for
use in the text. The file is fairly well commented, so if you’re a hacker, it might
contain everything you need to understand what’s going on.
All of the text files, graphics files, etc. are read into some main file. I call the
main file master.tex. In the future there might be different versions of this file,
each producing a different version of this text, as described in the paragraph
above for contributors. The files are called in with \include, and can thus be
turned off with \includeonly, though \includeonly is best used for editing,
not for producing a finished alternate version.
For example, suppose master.tex looked like figure 1. Then four files are
included. If we added a command \includeonly{rules} in the preamble, then
only one file, rules will be included. Similarly, one can comment out using %,
any file that one doesn’t want to include. The beginning of the real master.tex
file looks a lot more like the example shown in figure 2. As you can see from that
figure, I have chosen to make “fine-grained” sub-files; this allows the greatest
flexibility in turning them on and off and in rearranging them. The cost is of
course having to keep track of a couple hundred files. (Again, if you don’t like
having so many files, you’re welcome to take this collection, merge them into
one file, and distribute your version on the web.)
Already one can see that a huge number of Calculus texts could be created
by manipulating the included text files. However, the Free Speech Calculus Text
is far more flexible than this.
3.2
Pseudo-paragraphs
For these notes to have greater flexibility to hide/show material than would be
possible by using \includeonly and % on text files, the material in the files
must be chopped into many pieces. These pieces are defined by environments I
§3: How a teacher or serious user uses the text
Figure 2: Beginning of a real master.tex file
\documentclass{book}
\usepackage{free_calc}
\title{\huge\bf The Free Speech Calculus Text}
\author{Various authors, Gnu Free Documentation License}
\date{November 22, 2004}
\begin{document}
\section*{Introduction}
\include{text_files/introduction_to_whole}
\chapter{Background}
\include{text_files/introduction_to_background}
\section{Positive, negative numbers and inequalities}
\include{text_files/basics_about_numbers}
\include{text_files/inequalities}
\include{text_files/rules_of_basic_algebra}
\include{text_files/interval_notation}
\begin{homework}
\include{exercises/inequalities}
\end{homework}
\section{Functions}
\include{text_files/what_is_a_function}
\include{text_files/list_of_basic_functions}
\include{text_files/function_notation}
\include{text_files/function_examples}
\include{text_files/combining_functions}
\include{text_files/domain_and_range}
\include{text_files/one_to_one_functions_and_inverses}
\include{text_files/mathematical_models}
\begin{homework}
\include{exercises/function_domains}
\include{exercises/functions_exercises}
\end{homework}
\section{Analytic geometry}
\include{text_files/basics_of_graphs}
\include{text_files/lines_and_circles}
\include{text_files/graphing_functions_without_calculus}
\begin{homework}
\include{exercises/lines_and_circles}
\end{homework}
\section{End of chapter problems}
\begin{homework}
\include{exercises/funny_problems}
\end{homework}
\chapter{Limits}
\section{Elementary limits}
6
§3: How a teacher or serious user uses the text
7
call pseudo-paragraphs. Here’s the list of pseudo-paragraphs at present:
axiom
comment
cor (prints Corollary)
definition
derivation
discussion
example
fact
lemma
notation
procedure
program
proof
rule (prints Rule)
strategy
theorem
test
It’s easy to add new pseudo-paragraphs, although I’ll wait to explain how
until later (in the subsection on how a contributor uses the text).
All the material in the text is contained in pseudo-paragraphs. Each pseudoparagraph has a list of key-values as an argument. It is the key-values which
control which pseudo-paragraphs are displayed, and which hide their contents.
For example, we might type something like
\begin{example}{author=duckworth,author=newton,uses=sin}
Let’s show that the derivative of $\sin(x)$ is $\cos(x)$.
Etc.
\end{example}
Then this pseudo-paragraph would be controlled by the key-values “author=duckworth”,
“author=newton”, “uses=sin”.
3.3
Controlling pseudo-paragraphs with \Show
For the user to control which pseudo-paragraphs are shown and which are hidden, one uses the \Show command. The argument to \Show consists of a boolean
statement which is built from the standard logical constructors or, not, and,
(, ), and from the pseudo-paragraph names and key-values. Each pseudoparagraph which follows that \Show statement will be displayed if and only if
it satisfies the boolean statement defined by \Show. A \Show statement can be
entered anywhere in the document master.tex and it will affect the pseudoparagraphs which follow it. More than one \Show command can be entered;
each time it is entered the previous command is replaced entirely. Because I am
either not clever enough or do not have enough time, the items in the argument
to \Show must be separated by commas, which looks a little funny.
3.3.1
Examples of the Show command
For example, suppose one enters, the following \Show command in master.tex.
\Show{(,author=duckworth, and, author = newton,),or,uses=sin}
§3: How a teacher or serious user uses the text
8
Figure 3: Examples of pseudo-paragraphs
% 1
\begin{example}{author=duckworth,uses=limits}
Show that the limit of .... Etc.
\end{example}
%2
\begin{example}{author = newton, uses =sin}
Show that the derivative of $\sin(x)$ is \cos(x).
\end{example}
%3
\begin{lemma}
{author = wikibooks, author=duckworth, uses=e^x,uses=limits,style=formal}
The derivative of $e^x$ is $e^x$.
\end{lemma}
%4
\begin{discussion}{author=duckworth,author=newton}
Let’s look at what the previosu lemma means, etc.
\end{discussion}
Then each pseudo-paragraph will be shown if and only if it makes this boolean
statement true (a slightly more technical description of how this works is given
below). For the list of environments in figure 3 the ones which will be shown
are #2 and #4. If one entered
\Show{not,author=duckworth}
then only #2 would show. Note that in this case any key-value of the form
author = newton is simply ignored.
In addition to key-values, names of pseudo-paragraphs may be given in a
\Show command. Thus, if one entered
\Show{example,or,author=newton}
then environments #1,2 and 4 from figure 3 would be shown.
The command \Show{} removes all restrictions and shows everything.
The \Show command can be entered many times in a document. A simplified
example is given in figure 4. In this example, the only derivative rules which
would be shown are those which do use sin(x), cos(x), tan(x) but which do not
use limits. All of the applications of derivatives would be shown, etc.
There are times when one wants to show almost all environments, but if a
certain key type is present to only show particular values of that key. For example, the key type style has been defined and is meant to have values of the
§3: How a teacher or serious user uses the text
9
Figure 4: Modified simple master.tex file
\documentclass{article}
\usepackage{free_calc}
\begin{document}
\chapter{Derivatives}
\Show{(,uses=sin,or,uses=cos,or,uses=tan,),and,not,uses=limits}
\include{rules}
\Show{}
\include{derivative_applications}
\chapter{Antiderivatives}
\Show{examples}
\include{rules}
\Show{uses=volume_by_rotation}
\include{integral_applicaitions}
\end{document}
form style = formal, style = informal, style = standard, or something
like this. Not all environments have anything defined for style. Now suppose you wanted to write a “standard” text, and thus not show things that
are too “formal” or “informal”. You cannot issue a command of the form
\Show{style=standard} because most environments have no style defined, and
thus most environments will not pass this test. Instead, the way to enter this is
\Show{not,(,style=informal,or,style=formal,)}
The syntax for such a command is not very good because it requires the user
to enter a list of all the alternatives to style = standard. In the future, it
would be desirable to have a variation on the arguments to the \Show command
so that \Show{foo=empty, or, foo=bar} where empty would be a technical
value meaning “not present”. Thus this command would mean something like
“Show if the key-type foo is not present or if it is present and it equals bar”.
3.3.2
Caveat: the Show command requires keys
There’s one caveat to all these uses of \Show: they require key-values to be
entered in the pseudo-paragraphs. At present this has been done for “author”
throughout the text. The first couple of chapters also have the “label” and
“uses” keys installed. In any but the functionality has been created to manipulate the keys, as described above. There is also a macro for adding new
keys.
Here’s the list of current key-value pairs as applied to pseudo-paragraphs:
§3: How a teacher or serious user uses the text
10
Figure 5: Examples of the label key
% 1
\begin{example}
{author=duckworth,uses=limits,label=example:limit_of_sin_over_x}
Show that the limit of .... Etc.
\end{example}
%2
\begin{example}
{author = newton, uses =sin, label=example:deriv_of_sin_is_cos}
Show that the derivative of $\sin(x)$ is \cos(x).
\end{example}
%3
\begin{theorem}
{author=duckworth, label=theorem:taylors_theorem,version=1}
Let $f(x)$ be a function whose $n$th derivative exists. Then the
power series defined by etc.
\end{theorem}
%4
\begin{theorem}
{author=duckworth, label=theorem:taylors_theorem,version=2}
Let $f(x)$ be any a function whose $n$th derivative exists.
the polynomial etc.
\end{theorem}
author
uses
style
importance
establishes
Define
requires
label
version
Most of these are just descriptors which can be exist to provide a link between
the \Show command and the pseudo-paragraphs. However the label executes
certain actions within the pseudo-paragraph.
3.3.3
Fine-tuning with label
The key-values described so far, author, uses, style, etc. are rather broad. By
themselves, one could not obtain perfect control over which pseudo-paragraphs
are displayed. The label key overcomes this difficulty. In principal each pseudoparagraph has an almost unique label key, as shown in figure 5.
§3: How a teacher or serious user uses the text
11
Thus, in addition to the \Show commands described above, one could issue
commands of the form
\Show{label=example:deriv_of_sin_is_cos}
This allows for complete fine-tuning of which pseudo-paragraphs are displayed.
In addition, the label key writes a standard LATEX label which can later be
referred to with \ref. Thus, using figure 5, one could later write “By Taylor’s
Theorem \ref{theorem:taylors_theorem}”. This use requires (1) that most
pseudo-paragraphs have a unique label but that (2) truly alternative version’s
of the same result get the same label, but different versions. This is so one can
automatically use \ref{theorem:taylors_theorem}, even if this use comes in
an entirely different chapter from the lines which display Taylor’s theorem.
3.3.4
Other arguments to Show
In addition to the key-values just listed, \Show can take the names of pseudoparagraphs (as mentioned above), a list of which is given below.
There are also two arguments which are for editing purposes. If the word
files is given then each pseudo-paragraph which shows will display the name
of the tex file which contains that pseudo-paragraph. If the word keys is given
then each pseudo-paragraph which shows will display it’s list of keyvalues. Thus
\Show{keys,files,author=duckworth,and,author=newton}
will cause environment #4 in figure 3 to show, and it will print author =
duckworth , author = newton, as well as the name of the tex file which contained it.
3.3.5
A special psuedo-paragraph: homework
The homework environment is a special case of a psuedo-paragraph. This environment is used to start a homework section. You use it like so:
\begin{homework}
\include{exercises/problems_on_quotient_rule}
\include{exercises/problems_on_product_rule}
\end{homework}
This environment responds to the \Show command like the other psuedo-paragraphs.
If you want to edit a certain batch of homework problems, then you should
use the \includeonly command. However, homework includes two additonal
files that you can’t see in source code just given: a begin_hw file and a end_hw
file. These contain the commands which format the exercises and execute an
enumerate list. Thus, if you want to use \includeonly with homework, you
should enter something like:
\includeonly{exercises/begin_hw, %
exercises/problems_on_quotient_rule, exercises/end_hw}
§4: How a Contributor uses the text
12
The reason the files begin_hw and end_hw were created were so that you can
type \includeonly{text_files/product_rule} and automatically not have
the enumerate environment executed by the homework environment.
4
How a Contributor uses the text
First of all, you are free to copy the entire set of notes to your computer and
keep all your contributions to yourself. However, if you choose to contribute to
the collection that I’m maintaining and sharing, that would be great.
As I mentioned above, there’s lots of room for contributions: of text material,
graphics, LATEX etc. Again, I will accept additive contributions with little
oversight. Because of the tools built in to this text for showing/hiding things,
I think there is little reason to remove material, although at present there are
redundancies and duplications which should be removed.
WARNING
Anything you contribute should be free from copyright infringement. The
simplest way to make sure this is the case is to write it yourself. You should
not assume that you can use anything from a book, from the web, or even
from someone’s “free notes” unless there is a very explicit statement about
the copyright, placing the work under the GFDL or in the public domain.
Information available on www.wikipedia.com and www.wikibooks.com is
covered by the GFDL and may be incorporated in these notes. In fact,
a significant part of this material came from the wikibooks website as it
existed in November of 2004.
So how should one contribute? The answer has two parts: what sort of
guidelines should be used for writing the stuff up, and what sort of guidelines
should be used for getting your stuff into the main body of the text.
4.1
Getting your stuff into the main distribution
At present we don’t have any fancy system for contributors; this isn’t hosted
on www.sourceforge.com, and we don’t have a CVS system for keeping track
of versions and contributors. So, send your contributions to me. (If you have
knowledge about running a CVS or about putting stuff on sourceforge and
think these tools would help the Free Speech Calculus Text, then contact me
and let’s talk.)
If you do send me something, please send me very specific instructions for
where it should go; include the filename, the line number, the environment it
follows etc.
If you have a small amount of text, you can just paste it into an email to me.
If you have a whole new pseudo-section send it to me as an email attachment.
If you’ve edited a file so much that it would be simpler for me to just replace
§4: How a Contributor uses the text
13
my copy with the one you’ve made, that’s ok too, just attach the whole file, and
again give me clear instructions.
I prefer line-endings to be in the UNIX format, though this is easily changed
by any decent text-editor. I do request (though not require) that you use hardwrap instead of soft-wrap. In other words, set your text editor so that it enters a
return when your text reaches the right hand edge of the window. If you send me
material that has not been wrapped in this fashion, I will (1) re-wrap it myself,
and (2) communicate with you directly to find out if you didn’t understand
what I was asking for or if you have some reason not to do it this way.
If you’ve edited free_calc.sty much (i.e. if you’re improving my LATEX
code) then of course test your code as much as possible before sending it to me.
I’ll test it too at my end.
4.2
Figuring out where to edit
Some TEX set-ups have what’s called “reverse synchronization”. This means
that when you click on the dvi/pdf output, it sends you to your editor at the
place which produced the output you clicked on.
If you don’t have this (or haven’t bothered setting it up), I’ve included something to help you. The command \Show can take arguments of “keys” and/or
“files”. The effect of this command will apply to all the pseudo-paragraphs
which come after it; thus, you would usually put it in the pre-amble, but this is
up to you.
When executed, this will show information which will help you figure out
where and what to edit. \Show{keys} will cause each pseudo-paragraph to show
the key-values in the print-out. This way you can see, for example, that a proof
is by duckworth, and you will know to hide duckworth, if you want to get rid
of it.
\Show{files} will cause each pseudo-paragraph to show which text-file it
is contained in. This way, you will know where to go to edit it.
4.3
What to write
Well, mostly I figure you should contribute anything you like. If you’ve got a
favorite example, put it in. If you’ve got some notes that you’ve typed up, chop
them into pseudo-paragraphs, and put them in. If you see something missing,
put it in.
Here’s some big things that I know are missing.
Key-values. As described above the \Show command relies upon key-values being entered for each pseudo-paragraph. At present all pseudo-paragraphs
have values entered for author. The material in the Background and
Limits chapters has been somewhat thoroughly processed, but the rest of
the material has not been. In particular, each pseudo-paragraph should
have an almost unique label key added. This creates a standard LATEX
label, as well as provides a key for fine-tuning the \Show command. These
§4: How a Contributor uses the text
14
labels should be somewhat lengthy and descriptive, rather than label =
ex1523.
History. I think it would be nice to have short little bios for the major figures
in Calculus. The more entertaining the better. Don’t worry about them
being too long, anyone who doesn’t want them can turn it off!
Graphics. We need lots more graphics. It might be reasonable to aim for one
graphic per page. Don’t worry about having too many graphics, anyone
who doesn’t want so many can turn them off!
Exercises. We need lots of exercises. Either with or without solutions and
answers. (See also the “hacker” goal below for implementing code for the
homework.)
Alternative Calculus Implement a master.tex document to make alternative books. For instance, it would be nice to have master_infinitesimal.tex
which makes the book take an infinitesimal approach to Calculus. There is
also a “limits free” approach to Calculus, and it could have a master_no_limits.tex
file.
4.4
Writing stuff up
I don’t want to crimp anyone’s style, but, for the sake of working on a common text, we should have some consistency. Unfortunately, “some consistency”
means that it might take a little more work than if you were typing up your
own test or notes, but I hope that “little” is accurate.
4.4.1
Pseudo-paragraphs
Everything you contribute should be in smallish pieces which I’ve called pseudoparagraphs. At present most of the pseudo-paragraphs don’t have extensive
key-values, but it might be nice if you included at least the author, if it uses
any important functions or techniques, and of course the label key.
I understand, and agree, that the use of pseudo-paragraphs gives the text
an unfortunate chopped-up feel. This can be moderated later by turning off
numbering and titles for some of the environments, like comments or discussion. Then these environments would blend in with the background. But I
feel that using the pseudo-paragraphs is the best way to allow the mixing of
different examples, theorems, etc, by different authors, and allow these things
to shown/hidden.
4.4.2
Creating new pseudo-paragraph environments and new keyvalues
Although there’s a pretty large list at present of pseudo-paragraph environments
and key-values, there will undoubtedly be someone who wants to create new
§4: How a Contributor uses the text
15
ones. This makes sense, but use restraint as the whole package could become
unwieldy if the lists become really large.
To create a new key-value called foo you would enter
\NewPPKey{foo}
After this has been done then any pseudo-paragraph could have entries like the
following:
\begin{example}{author= duckworth, foo = bar}
Etc
\end{exmaple}
\end{verabatim}
To create a new psuedo-paragraph called {\tt foo} you would enter two
commands thus
\begin{verbatim}
\newtheorem{innerfoo}{\footitle}
\NewPP{foo}{Foo}{\thingy}
You have to create an environment called innerfoo; this doesn’t have to be
done with \newtheorem, but the amsthm package makes the \newtheorem powerful and convenient for our purposes. Thus, you can decide how the counter for
foo should work by entering \newtheorem{innerfoo}{\footitle}[section]
or \newtheorem{innerfoo}[example]{\footitle}.
The command \footitle is defined by the \NewPP command. For this
to work you must use \footitle. The default for \footitle is the second
argument to the \NewPP command, in this case Foo. You can re-define this as
an optional argument when you use foo. Thus
\begin{foo}[Newton’s Foo]{author = duckworth}
would make that particular foo environment have the title Newton’s Foo.
The command \thingy defines an end of pseudo-paragraph symbol. For
proofs, this is the amsmath command \qed which makes the box
. You can
leave this symbol empty, as is the case for lemmas, theorems and corollaries. An
illustration of a generic symbol to mark the end of examples, rules, definitions
etc. is a horizontal line:
The command to create this line has been called \linesep. Here’s the actual
code I used to create the example environment:
\newtheorem{innerexample}{\textcolor{blue}{\exampletitle}}[section]
\NewPP{example}{Example}{\linesep}
The command \textcolor{blue}{\exampletitle} makes the Example appear in blue. The command [section] tells LATEX to number examples within
sections, so that the numbering restarts with a new section.
§4: How a Contributor uses the text
4.4.3
16
Pseudo-sections
Pseudo-sections are what I call the files that are included into master.tex.
What should form a pseudo-section? A pseudo-section should be a cohesive
part of a single section. Obviously, this is a little vague, or, if you like, a matter
of preference. One way to figure out what should be a pseudo-section is if you
can imagine someone wanting to hide/show/move the whole body of material.
So, the quotient rule and the product rule could each form a pseudo-section;
you would probably include them both in a single real section. Similarly, in the
real section of integrating rational functions, you might include a pseudo-section
for each of the following: polynomial division, completing the square, partial
fractions, etc.
4.4.4
Notation
Of course we want the text to have somewhat consistent notation. Of course all
Calculus books use more than one notation for derivatives, and we will follow
this lead.
There are two ingredients in creating a moderate amount of consistency. (1)
Don’t be too idiosyncratic, or too technical,
or too perfect
etc. So, in Calculus I,
R
R
d
instead of Dx . We’ll write x2 dx, not x2 dx. We’ll write out “for
we’ll use dx
all x in [0, 1]” instead of “∀x ∈ [0, 1]”. (2) Use mark-up language for things
that have specific meaning, especially things that have specific meaning that
someone else might write differently. Thus use the command \ddx which means
“the derivative with respect to x”, instead of writing \frac d{dx}. Why the
d
command? Well, dx
has specific meaning, it’s not just a fraction with a d on top
and d and x on the bottom. Plus, if we all use \ddx then all of our derivatives
look the same. Moreover, if one of us is crazy enough to want their derivatives
d
then all they have to do is re-define the command
to look like Dx , Dx , or dx
\ddx, and presto, all of the derivatives have changed their appearance.
It would be nice, but I doubt it’ll happen, if we all used \sin{x} and
\arcsin{x}. That way, by changing a single definition at the top, those who
like to write sin x could have all their sins look this way and those who like to
write sin(x) could have all their sins look this way. Similarly \arcsin{x} could
be defined to produce sin−1 (x) or sin−1 x, or arcsin x, or arcsin(x) according to
the teacher user’s whim. I suspect that such notational enforcement will be too
difficult to maintain.
Brows the file free_calc.sty for a list of built in notations, and feel free to
add some.
4.5
Graphics
Of course LATEX can import a huge variety of graphics. The poorest quality ones
are bitmaps like JPEG, PNG or GIF. The nicest looking ones are in PostScript
or PDF, or generated internally by packages like eepic.
§4: How a Contributor uses the text
4.5.1
17
Formats and openness
The fact that this text is licensed under the GFDL, that it will be used by and
contributed to by a wide variety of people, and that it will have a wide variety
of applications (i.e. text, electronic, web-based, etc), means that the format and
approach to graphics might be different than what you do for your tests/lecture
notes, etc. Here are further considerations of these points.
• The GFDL requires that any distribution of this text also provide source
files for the text and the graphics. Thus, if you make a plot in Maple,
there should be a Maple source file somewhere that creates that plot.
• Since many people might be looking at the source and modifying it, it
would polite and nice to have that source be used by a program that is
widely available, preferably on different platforms.
• Since many people might be looking at the source and modifying it,
it would be nice and polite if that source is human-readable, not binary/cryptic XML. Thus, if you use Maple it is preferable to have the
Maple code for a plot saved as plain-text, rather than as a worksheet.
• Since different uses might be made of this text, it is best to use tools which
can export graphics into a wide variety of formats.
• If someone wants a particular application of this text that requires a particular graphic format, it would be nice if changing the graphics was reasonably easy. Since there will be hundreds of graphics, this means that
the graphics should be done in such a way to allow batch-processing. In
other words, there should be a small set of commands which would allow
all the graphics to be re-generated. For Maple this is most easily done by
creating a single source code file with the the plot-settings at the top, and
all the plots afterwards. This has been done in the plot_library.mpl
file within the graphics_maple directory. For Xfig all the fig files can be
processed from the command line with the fig2dev command.
The above items list recommended procedures. Here’s some recommended
tools: Maple (expensive), Xfig (free), Mathematica (expensive), Gnuplot (free).
4.5.2
Some other graphics tools
I’ll mention here some other tools for producing graphics to use in LATEX. I’ve
used some of them, but not all, and in case this is entended to just be a list.
• XY-pic. This is a TEX macro package that excels at drawing mathematical
diagrams. “Diagrams” are pictures that consist mostly of nodes and things
connecting the nodes. The nodes can be simple circles, formulas, points,
etc. The things connecting the nodes can be lines, arrows, curves, etc.
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18
• PS-tricks. This a TEX macro package that gives access to PostScript
commands within a TEX environment. The syntax is geared to creating
the sort of drawings that are typically useful in mathematics.
• potrace. This is a program that can transform a black and white bitmap
image (like JPEG, GIF, PNG, etc) into PostScript or PDF, and even X-fig
formats.
• pstoedit. This is a program that tranlates PostScript and PDF into other
editable formats.
4.5.3
Graphic size
Of course it’s a matter of opinion as to what size a graphic should be, and this
size depends on where it’s going to be placed, as well as how complicated the
picture is. Here’s some rule’s of thumb.
To put a graphic in the margin the width should be 5cm. To put a graphic
in a displayed figure, the size should be larger, but it’s not clear by how much.
As a rule, when people first make graphics they tend to make them too big. A
width of 7cm looks fine for most displayed graphics, and probably it should be
at most 10cm.
4.5.4
Graphics commands
At present there are small number of custom graphics commands. The command
\margingraph{filename}{caption}{label}
creates a graphic figure in the margin. It gets the graphic from the file filename,
it gives it a caption caption and it makes a LATEX label label. This command
at present assumes that the size of the graphic will fit into the margin; although
I see no reason not to have it automatically size the graphic to width of the
margin.
If one wants to put a graphic in the margin without using the preceding
command, one can type
\marginpar{\centering\includegraphics{filename}}
If one wants a caption in the margin graphic, one can use the command \figcaption
within the \marginpar.
The free_calc.sty package loads the graphicx package, so all of the commands described in that package can be accessed. However, the internal command \Ginclude@graphics has been redefined so that when a \Show{files}
command has been issued, the name of the graphics file will be printed next to
the graphic. This change should not affect the user at all (unless I’ve made a
mistake in the implementation).
§4: How a Contributor uses the text
4.5.5
19
Future graphics procedures
At present, all of the graphics have been imported in PDF. This looks quite nice,
but there are a few features which make it less than ideal. (1) The fonts come
from the program which generate the graphic, rather than LATEX. (2) Because
of this one cannot have formulas. (3) If one resizes the graphic then the lines,
fonts, etc are also resized, resulting in lines and fonts that are too big/small.
All of these problems can be avoided, and in the future probably will be.
Firstly, if one uses PostScript then the fonts from the graphic generating program can be replaced with LATEX fonts. The package psfrag.sty does this,
although it can be done in other ways as well. This solution combines the
full power of the graphic generating programs with the nice features of LATEX.
However, re-sizing graphics still changes the thickness of lines, so the original
graphic needs to be close to the size that will be finally used. Also, this solution
introduces an extra step into the graphic process: first one creates the graphic,
then one exports, then one does something to replace the original fonts with
LATEX fonts.
Another solution involves using the package eepic. This package greatly
enhances the LATEX picture drawing tools. The program Xfig can export directly to eepic commands which are contained within a LATEX environment.
This means that the fonts are automatically correct, and that one can edit the
file as a LATEX file in order to position formulas, use macros etc, and that resizing will not affect the lines, fonts etc, as these are generated by LATEX. The
program Gnuplot can also export its plots as eepic commands. Finally, Maple
can export plots as a list of points, and the package coordsys has a facility
for importing these points into a eepic environment. The eepic solution has
two drawbacks: (1) eepic cannot reproduce all of the features of Xfig so some
drawings cannot be fully exported to eepic, (2) if one uses Maple to export
points, which are then combined with coordsys and epic, and if your plot has
more than one function, a separate graphics file is created for each function (I
suspect that this feature can be changed in a future version of the tt coordsys
package).
I have not yet implemented (or even experimented with) these solutions,
but I think that in the future this text will probably adopt one or more of
the procedures just described. As mentioned elsewhere, the final format of this
text should be in PDF. Thus, all of the procedures just described would require
processing the LATEX source as follows:
LATEX source −→ dvi −→ PostScript −→ PDF
This creates an extra couple of steps as compared to using pdf-LATEX, however
the psfrag and eepic packages are not compatible with pdf-LATEX and are not
likely to become compatible in the near future. (The extra processing steps
required by these packages fits in well with the multi-step procedure involving
dvi and PostScript. In contrast, when making a PDF document directly, each
page needs to be completely finished and processed at the time the PDF is
created.)
§5: How a hacker uses the text
4.6
20
Some re-defined LATEX commands
I’m sorry, but I’ve redefined some commands from standard LATEX and other
packages. In general this is a bad idea, but I will keep my changes unless
someone can convince me that these particular cases are bad. As far as I can
tell, this shouldn’t cause problems, but you never know, so here’s a list of them.
• \rule. In LATEX this creates a black rectangle. However, I wanted to use
“rule” as the name of an environment for giving a Rule, like the rule for
a derivative. The original LATEX command has been saved with the new
name \inkrule. Thus \inkrule{.5in}{1in} will make a black rectangle
.5in by 1in in size.
• \begin{proof} from the amsthm package has been re-created. The new
proof environment should work the same way as the old one.
• \includegraphics from the graphicx package has been re-defined so
that if \Show{files} is given then the name of each graphic file will
be displayed next to the graphic. In other respects the new version of
\includegraphics should work the same as the old one.
• \include has been re-defined so that it does not issue \clearpage commands and so that LATEX keeps track of which file is currently being processed. In other respects the new version of \include should work the
same as the old one.
• \begin{comment} from the verbatim package has been re-defined so that
it creates an environment which prints Comment which presumably contains comments from the author to the reader. The original command has
been saved with the new name HiddenComment. Thus
\begin{HiddenComment}
Put some stuff here.
And more stuff here.
\end{HiddenComment}
will hide all of the material enclosed in the environment.
• \l and \r are now synonyms for \left and \right respectively.
• \i and \j now produce î and ĵ.
5
How a hacker uses the text
This section contains specific goals which involve LATEX coding, or software
coding in the usual sense. Of course, you may see other things that you would
rather hack, in which case go for it. But, if you’re interested in things that I
think need fixing, here they are.
§5: How a hacker uses the text
21
• I suspect that a knowledgeable LATEX programmer would see that much
of the LATEX code I’ve created in free_calc.sty is poorly done, so suggestions and fixes in this regard would be welcome.
• The hyperlinks have not been set up at all. This should be a relatively
straight forward project for someone who’s interested.
• Various code issuues with the homework.
– General layout issues for the homework, described in the next major
item. Also, need to change where directions for a set of exercises
appears.
– Each exercise should be given in three parts: the problem, a hint,
and a solution. Then various options would be set for which parts
should display.
– In addition, it would be nice to have key-values for homework. This
would not necessarily be the same key-values that psuedo-paragraphs
use. Also, we probably wouldn’t want to enter key-values for every
problem, but rather sets of problems. So, maybe there would still be
an outer psuedo-paragraph like so:
\begin{problems}{uses = product_rule}{
\item $x\sin(x)$
\item $x^2e^x$
Etc
\end{problems}
Using this approach would make it easy to implement \Show{ keys,files}
as well.
• Layout. The notes at present do not look as good as most books. The
layout needs to be tuned. There should probably be more than one layout
option (i.e. a generic option, an e-book option, etc, stored in the layouts
directory). Here are a few particulars.
– We need more distinct colors on most pages. Note: we don’t need
too many colors, and we don’t need them just for entertainment.
However, a typical calculus book will have colors which highlight
examples, theorems, rules etc. Also, a few colors on every page is
more pleasing to the eye than pure black and white.
– The exercise pages should have a different layout. They should have
smaller margins, smaller fonts, and be in twocolumn mode.
– The numbering of the examples, definitions etc. should be tweaked.
In particular, all of these numbers should be reset at the beginning
of each section, but this should not use the default LATEX scheme of
numbering as “Example 2.3.11” for chapter 2, section 3, example 11.
Rather, it should just be “Example 11”, but this should get reset at
the beginning of the section.
§5: How a hacker uses the text
22
• Improve the syntax for the Boolean constructions in the \Show command.
In particular, the requirement of commas everywhere is a bit unnatural
and likely to lead to user errors.
• Print out a warning message when someone enters the same “label” key
twice. Note that this should be a gentle warning, since there are times
where we want the same label to appear twice, at least in editing mode,
if not in a final print-out.
• Implement cross-checking. So, if an environment requires limits, it should
check if limits have been defined. If an example requires Taylor’s Theorem,
it should check if Taylor’s Theorem has been stated.
• Implement importance levels. Thus, each example could be rated “importance=1”, “importance=2”, etc. That way, the user could decide to show
only the really important stuff.
Using levels would require discipline. We don’t want twenty levels, and
everyone will disagree over what’s the most important thing. Still, might
as well put the option in there.
• Modify the graphics input commands to make them easier and more consistent for the average user.
In addition, modify the command which shows the name of the graphic
file, so that it shows the full path-name, i.e. it shows the name of the
directory that the file is contained in.
• Using the unique name-labels, implement commands to show/hide “from...to”.
Thus, one could say
\Show{from=example_limit_of_sin_over_x,
to=example_of_e_to_the_x_over_x}
and this would show those pseudo-paragraphs between the two that are
specified by the unique labels.
• Tweak the \Show command so that formatting comes out better. Here
are some problems with the currrent set-up. Suppose you want to show
only theorems. Then the final product will have many empty pages which
were generated by \chapter and \section commands which didn’t have
any theorems in them.
The obvious way to fix this is with flags; the psuedo-paragraphs could
set a flag which says “this section is not empty”, the section and chapter
commands would check to see if the section is not empty before displaying
the heading. Since the section and chapter commands come before the
psuedo-paragraphs, the flag will have to be stored in a .aux file, which
will record the result from the previous run; the default should be to show.
§5: How a hacker uses the text
23
Thus, if you are showing only theorems, and if chapter 1 has no theorems,
on the first run the chapter and section commands would display, this
would be followed by empty pages. But on the second run, LATEX would
check the .aux file, would see that these sections were empty, and the
section and chapter command would not produce anything: it would just
gobble it’s contents.
A downside of this is that one might have to run LATEX three times (twice
to get rid of the blank pages, and then one more time to make sure the
references have stabilized).
• Investigate running this text on sourceforge, or as a CVS, or even as a
Wiki.
I’ve been hesitant to set it up as a Wiki so far, for the following reasons.
Firstly, Wikis are presently set up to edit and display documents in html,
whereas this project is in LATEX. Without the use of html, it seems to
me that there’s no advantage of using a Wiki as compared to some other
system with fairly free access. For example, if we set up a CVS and make
it really easy for people to have accounts on it, this might be a good thing.
Secondly, in the early stages I think this project needs more structure than
is usually present in a Wiki. In other words, when the goals are being
defined and the framework for adding material is changing, it seems best
to me to have an editor/coordinator. Finally, this project involves coding,
and code does not lend itself to anonymous/random contributions. There
are two levels of code here: the internal code that makes things work, and
the user-level code that is used in writing up the material. Changes in the
internal code need to be debugged and evaluated before being distributed.
As far as the user code goes, I think this could mostly be open to anyone
making changes at any time, except for the issue of debugging. I think
any change should pass some sort of a test designed to find errors.
• Investigate other markup languanges or software packages for the whole
text. Here’s a description of one possibility that I’m aware of.
The language ConTEXt looks like a really advanced approach to automatic
typsetting. This package is built on top of TEX, using PDF; one of the
main contributors was Hans The Than, the person who invented pdf-TEX.
It can be thought of as a parallel to LATEX: it consists of a large number
of macros which describe abstract markup, as opposed to TEX’s detail
oriented markup.
ConTEXtis designed to create dynamic documents and address certain deficiencies in LATEX. The word “dynamic” is used technically, as in contrast
to “static”. Most documents that you write by hand are static, they will
not change except when you change them, and this is probably not often
compared to the number of times they will (hopefully) be used. In contrast dynamic documents might change for every user, or might change
every day, or might change while you’re using them.
§5: How a hacker uses the text
24
These calculus notes represent an attempt to make LATEX create dynamic
documents in a limited way; it has required adding on a lot of packages
and code whereas ConTEXt was created from the ground-up to do this sort
of thing. ConTEXt can create PDF documents which have interaction; i.e.
buttons, forms, conditional execution etc. One drawback of this capabality
however, is that the resulting PDF document can not be run on many PDF
viewers; it should probably be run on Adobe Acrobat Reader, though in
the future other readers may become powerful enough to implement all of
these features as well.
Another drawback of ConTEXt is that fewer people know about it, and that
the documentation isn’t as complete as it is for LATEX. Here’s some links
to find out more about ConTEXt.
The home-site of the inventors of ConTEXt:
http://www.pragma-ade.com/
The show-case manual (you probably need Adobe Acrobat Reader for this,
as opposed to X-pdf, TeXShop, Apple Preview, GSview, Foxit, etc, you
also need to download additional filess if you want to follow the links)
http://www.pragma-ade.com/show-man.pdf
Here’s an useful website which discusses how to migrate LATEX to ConTEXt:
http://www.berenddeboer.net/tex/
• I think it should be fairly easy to put together a simple graphical interface
to many of the options in this text.
All that would be needed is to take some graphical scripting language, ask
some questions, write the responses to a configuration file, and then to
run pdf-LATEX in the background. All of this might be possible directly
through ConTEXt, however I think it might be best at present to not commit
to moving everything to ConTEXt.
Ultimately, I imagine having a full package on a CD that would work on
any computer. The student version of the CD would probably have the
broad outline set-up by the teacher. Then the student could go home,
start with the “standard” version of the class text, and show only the
examples, show extra examples, show extra graphs, etc. An important
possibility here is to get hints on the exercises.